Curve generating method using tangent vectors

ABSTRACT

The method of the invention includes determining a circular arc (CAR) passing through three discretely given consecutive points (P i-1 , P i , P i+1 ), determining a tangent vector of a tangent line contacting the circular arc (CAR) at the central point (P i ) of these three points, thereafter performing a spline interpolation between the two points P i-1 , P i  using position vectors and tangent vectors at the points (P i-1 , P i ), thereby obtaining a curve smoothly connecting the two points P i-1 , P i , and thereafter determining a point sequence connecting curve (CVL) by similarly performing an interpolation between every two adjacent consecutive points.

BACKGROUND OF THE INVENTION

1. Field of the Invention

This invention relates to a curve generating method and, more particularly, to a method of generating a three-dimensional curve which becomes necessary when generating a curved surface.

2. Description of the Related Art

A curved surface of a three-dimensional metal mold or the like on a design drawing is generally expressed by a plurality of section curves, but no profile data is shown for the shape of the area lying between a certain section curve and the next adjacent section curve. In numerically controlled machining it is essential that machining be carried out so as to smoothly connect these two section curves despite the fact that the profile between them is not given In other words, this means that machining must be performed by generating the curved surface between the two section curves from such data as that indicative of the section curves, recording on an NC tape the data concerning the generated curved surface, and carrying out machining in accordance with commands from the NC tape. To this end, there has been proposed a method of generating the curved surface of a three-dimensional curved body in accordance with predetermined rules using data (e.g. section curves and the like) specifying the three-dimensional curved body

FIGS. 5(a)-(d) for describing a curved surface generating method, in which a curved surface CS [see FIG. 5(c)]is generated by providing three-dimensional curves (reference curves) 11a, 11b [see FIG. 5(a)]of a curved surface cut by a predetermined section, dividing each of the reference curves 11a, 11b into N equal segments [see FIG. 5(b)], and connecting corresponding ones of the partitioning points by straight lines

In this curved surface generating method, the reference curves 11a, 11b, which are the three-dimensional curves, must be specified. To this end, a sequence of discrete points P_(li) (x_(i),y_(i),z_(i)) (i=1,2, . . . ) is given with regard to the reference curve 11a, as shown in FIG. 5(d), a sequence of discrete points P_(2j) (x_(j),y_(j),z_(j)) (j=1,2,...) is given with regard to the reference curve 11b. and curves (reference curves) connecting these point sequences are obtained by performing interpolation between points so as to smoothly connect the respective point sequences.

In this conventional method of generating the point sequence connecting curves, it is necessary to determine a tangent vector at each point However, the method of determining these tangent vectors is a major undertaking requiring matrix computations, inverse matrix computations, etc., and it is impossible for an ordinary curved surface generating apparatus on the personal computer level to determine the tangent vectors.

SUMMARY OF THE INVENTION

Accordingly, an object of the present invention is to provide a curve generating method through which tangent vectors can be determined in a simple manner, thus making it possible to simply obtain curves smoothly connecting point sequences.

The method of the present invention includes determining a circular arc passing through three discretely given consecutive points P_(i-1), P_(i), P_(i+1), determining a tangent vector of a tangent line contacting the circular arc at the central point P_(i) of these three points, thereafter performing a spline interpolation between the two points P_(i-1), P_(i) using position vectors and tangent vectors at the points P_(i-1), P_(i), thereby obtaining a curve smoothly connecting the two points P_(i-1), P_(i), and thereafter determining a point sequence connecting curve by similarly performing an interpolation between every two consecutive points discretely given.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a view for describing the general features of the present invention;

FIG. 2 is a block diagram of an apparatus for practicing the present invention;

FIG. 3 is a flowchart of processing according to the present invention;

FIG. 4 is a flowchart of different processing according to the present invention; and

FIGS. 5(a)-5(d) for describing a method of generating a curved surface.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT

FIG. 1 is a view for describing the general features of the present invention.

P_(i) (i=1, 2,... n) represents a sequence of points, CVL a curve smoothly connecting the sequence of points, CAR a circular arc passing through three consecutive points, C_(i) the linear distance between adjacent points, and S_(i) (t) the coordinates of a point on the curve CVL.

The circular arc CAR passing through three discretely given consecutive points P_(i-l), R_(i), R_(i+1) is determined, followed by determining a tangent vector T_(i) of a tangent contacting the circular arc at the central point P_(i). A tangent vector T_(i-l) at the point P_(i-1) is then found in similar fashion.

Next, by using position vectors P_(i-l), P_(i) and tangent vector T_(i-l), T_(i) at the points P_(i-1), P_(i), a spline interpolation is performed between the two points P_(i-1), P_(i) to obtain the coordinates S_(i) (t) (where t is a value increased at increments of 0.1 from 0 to 1), and a curve which smoothly connects the two points P_(i), P_(i+1) is determined.

Thereafter, and in similar fashion, the point sequence connecting curve CVL is obtained by performing an interpolation between every two consecutive points of the discretely given points invention, and FIG. 3 is a flowchart of processing In FIG. 2, numeral 201 denotes a keyboard for data input, 202 a processor, 203 a ROM for storing a control program, 204 a RAM, 205 a working memory, 206 a curve/curved surface memory for storing generated curve and curved surface data, 207 an output unit for outputting generated curved surface data to an external storage medium 208 such as a paper tape or magnetic tape, 209 an address bus, and 210 a data bus

Processing for generating a curve in accordance with the invention will now be described in accordance with the flowchart of FIG. 3.

(a) First, data specifying a three-dimensional curved surface, e.g. a point sequence (a position vector P_(i) at each point) specifying the curve CVL (FIG. 1), is inputted from the keyboard 201.

(b) Next, the processor performs the operation 1 →1.

(c) Thereafter, the circular arc CAR (see FIG. 1) passing through the discretely given three consecutive points P_(i-1), P_(i), P_(i+1) is found.

(d) When the circular arc CAR has been found, a unit tangent vector P_(i) of a tangent line contacting the circular arc CAR at the central point P_(i) is obtained.

(e) Next, the linear distance C_(i) between the point P_(i-1) and the point P_(i) is computed.

(f) When the distance C_(i) has been found, the coordinates S_(i) (t) of points on the curve smoothly connecting the points P_(i-1), P_(i) pare computed by varying t over a range of from 0 to 1, namely by performing spline interpolation, in accordance with the following equation: ##EQU1##

In Eq. (1), S_(i) (t) represents the coordinates of point P_(i-1) if t=0, the coordinates of point P_(i) if t=1, and the coordinates of the midpoint if t=0.5. Accordingly, nine points between points P_(i-l), P_(i) are obtained by incrementing t at 0.1, interpolation is performed over this interval, and the coordinates of each point are stored in the memory 206. It should be noted that Eq. (1) is an induction equation of a Ferguson curve segment.

(g) Next, it is determined whether the point P_(i) is the end point of the curve If P_(i) is the end point, & curve generation processing is terminated

(h) If the point P_(i) is not the end point of the curve, however, i is incremented by the operation i+1→i and processing is repeated from step (c) onward to perform interpolation between every two consecutive points discretely given, whereby a point sequence connecting curve is obtained.

The processing of FIG. 3 is for a case where a tangent vector is found using three consecutive points However, a tangent vector can also be obtained by using five consecutive points. FIG. 4 is a flowchart for obtaining a tangent vector using five points, and obtaining a curve using the tangent vector.

(a) First, data specifying a three-dimensional curved surface, e.g. a point sequence (a position vector P_(i) at each point) specifying a curve is inputted from the keyboard 201.

(b) Thereafter, a circular arc passing through three discretely given consecutive points P_(i-2), P_(i-1), P_(i) is found.

(c) When the circular arc has been found, a unit tangent vector e_(l) of a tangent line contacting the circular arc at a point P_(i) at the right end is obtained

(d) Next, a circular arc passing through the three discretely given consecutive points P_(i-1), P_(i), P_(i+1) is found.

(e) When the circular arc has been found, a unit tangent vector e₂ of a tangent line contacting the circular arc at the central point P_(i) is obtained.

(f) Next, a circular arc passing through the three discretely given consecutive points P_(i), P_(i+1), P_(i+2) is found.

(g) When the circular arc has been found, a unit tangent vector e₃ of a tangent line contacting the circular arc at a point P_(i) at the left end is obtained

(h) When the unit tangent vector e₁ -e₃ have been obtained, a unit tangent vector P_(i) ' at the point P_(i) is found in accordance with the following equation:

    P.sub.i '=w.sub.1 ·e.sub.1 +w.sub.2 ·e.sub.2 +w.sub.3 ·e.sub.3                                         (2)

In the above wi (i=1, 2, 3) is a weighting coefficient obtained in accordance with the following equation so that the unit tangent vector where the interval between points is small will be reflected in the curve:

    w.sub.1 =1-(C.sub.i +C.sub.i-1)/(C.sub.i-1 +2C.sub.i +2C.sub.i+1 +C.sub.i+2)

    w.sub.2 =1-(C.sub.i +C.sub.i+1)/(C.sub.i-1 +2C.sub.i+ 2C.sub.i+1 +C.sub.i+2)

    w.sub.3 =1-(C.sub.i+1 +C.sub.i+2)/(C.sub.i-1 +2C.sub.i +2C.sub.i+1 +C.sub.i+2)

where the distance between points P_(i-2), P_(i-1) is C_(i-1), the distance between points P_(i-1), P_(i) is C_(i), the distance between points P_(i), P_(i+1) is C_(i+1), and the distance between points P_(i+1), P_(i+2) is C_(i+2).

(i) Next, the tangent vector T_(i) is computed in accordance with the equation

    T.sub.i =C.sub.i P.sub.i '

using the linear distance C_(i) between the point P_(i-1) and the point P_(i).

(j) This is followed by varying t over the range of from 0 to 1 in accordance with Eq. (1) to compute the coordinates S_(i) (t) of points on the curve smoothly connecting the points P_(i-1), p_(i).

(k) Next, it is determined whether the point P_(i) is the end point of the curve. If P_(i) is the end point, curve generation processing is terminated.

(m) If the point P_(i) is not the end point of the curve, however, i is incremented by the operation i+1 →i and processing is repeated from step (b) onward to perform interpolation between every two consecutive points discretely given, whereby a point sequence connecting curve is obtained.

In accordance with the present invention, the arrangement is such that a tangent vector at each point of a point sequence is obtained by using three or five consecutive points. As a result, tangent vectors can be obtained in a simple manner and, hence, a curve smoothly connecting the point sequence can be obtained in simple fashion.

With the method of obtaining tangent vectors in accordance with Eq. (2) using five points, a curve can be minutely altered by changing the manner in which each weighting coefficient is decided, thereby making it possible to generate the desired curve. 

We claim:
 1. A method of machining a three-dimensional curved surface comprising the steps of:(a) detecting a circular arc passing through three discretely given consecutive points P_(i-1), P_(i), P_(i+1) of the curved surface; (b) determining a tangent vector of a tangent line contacting said circular arc at the central point Pi of said three points; (c) determining a curve smoothly connecting the two points P_(i-1), P_(i) by performing a spline interpolation between the two points P_(i-1), P_(i) using position vectors P_(i-1), P_(i) at the points P_(i-1), P_(i) and tangent vectors T_(i-1), T_(i) at the points P_(i-1), P_(i) determined in step (b); (d) determining a point sequence connecting curve by performing a spline interpolation between every two adjacent consecutive points; and machining the three-dimensional curved surface according to the point sequence connecting curve.
 2. A method of machining according to claim 1, wherein a curve Si(t) determined in step (c) between the two points P_(i-1), P_(i) is expressed by the following equation; ##EQU2## and the spline interpolation is performed between the points P_(i-1), P_(i) by varying t over a range of from 0 to
 1. 3. A method of machining according to claim 1, wherein step (b) is performed by computing said tangent vector in accordance with the following equation:

    T.sub.i =C.sub.i P.sub.i,

where P_(i), represents a unit vector of a tangent line contacting the circular arc at the point P_(i), and C_(i) represents the distance between the two points P_(i-1), P_(i).
 4. A method of machining a three-dimensional curved surface comprising the steps of:(a) determining a circular arc having left and right ends and passing through three discretely given consecutive points P_(i-2), P_(i-1), P_(i) and determining a unit tangent vector e₁ of a tangent line contacting said circular arc at a point P_(i) on the right end; (b) determining a circular arc passing through three discretely given consecutive points P_(i-1), P_(i), P_(i+1) and determining a unit tangent vector e₂ of a tangent line contacting said circular arc at a central point P_(i) ; (c) determining a circular arc passing through three discretely given consecutive points P_(i), P_(i+1), P_(i+2) and determining a unit tangent vector e₃ of a tangent line contacting said circular arc at the point P_(i) on the left end; (d) increasing the weighting w_(i) (i=1-3) of a unit normal vector e_(i) where the distance between two points is short, and determining a unit tangent vector P_(i), at the point P_(i) in accordance with the equation

    P.sub.i '=w.sub.1 e.sub.1 +w.sub.2 e.sub.2 +w.sub.3 e.sub.3 ;

(e) computing a tangent vector in accordance with the following equation:

    T.sub.i =C.sub.i P.sub.i '

where C_(i) represents the distance between the two points P_(i-1), P_(i) ; (f) determining a curve smoothly connecting said two points P_(i-1), Pi by performing a spline interpolation between the two points, P_(i-1), P_(i) using tangent vectors P_(i-1), P_(i) at the points P_(i-1), P_(i) and position vectors T_(i-1), T_(i) at the points P_(i-1), P_(i) ; (g) determining a point sequence connecting curve by performing a spline interpolation between every two adjacent consecutive points; and matching the three-dimensional curved surface according to the point sequence connecting curve.
 5. A method of machining according to claim 4, characterized in that, in said sixth step, a curve S_(i) (t) between the two points P_(i-1), P_(i) is expressed by the following equation: ##EQU3## and the spline interpolation is performed between the points P_(i), P_(i+1) by varying t over a range of from 0 to
 1. 